INFORMAL LOGIC 

XIII.l, Winter 1991 

Quantifying Support 

JOHN BLACK Malaspina College 

A. Introduction 

Teachers of informal logic and critical 
thinking have come to be concerned in re-
cent years about the inadequacy of standard 
methods for assessing argument strength. 
The traditional notions of acceptability, 
relevance and sufficiency, while conceptual-
ly adequate, do not yield very fine-grained 
distinctions between arguments of varying 
strength. A number of philosophers have 
therefore initiated the search for a better 
system. 

This theme was pursued at the Second 
Conference of the International Society for 
the Study of Argumentation, held in June 
1990 at the University of Amsterdam. 
Stephen Thomas gave a presentation 
describing the system of adjectival grada-
tions (strong, moderate, weak etc.) he uses 
in teaching the evaluation of arguments. The 
system is an elaboration on the one he has 
published in Practical Reasoning in Natural 
Language. I Mark Battersby went further, 
at least in intent: he gave a presentation call-
ing for the investigation of quantitative 
methods for assessing degree of support, 
and hence the strength of arguments. He did 
not, however, provide details of a possible 
scheme for doing this, since his preliminary 
inquiries into its feasibility unearthed a 
number of difficulties. 

In this paper I shall argue that, to the 
extent that it is plausible, Battersby's pro-
gramme can be accomplished using a ver-
sion of the well-known probability calculus. 
I shall provide an explanation of some of 
the complexities involved in applying the 
calculus to this programme, and a solution 
to some of the difficulties discovered by 

Battersby. However, I shall also record my 
reservations about the generality of the ap-
plication of the calculus to the assessment 
of argument strength. 

B. The Overall Strategy 

In applying the probability calculus to 
argument assessment, I shall make the 
assumption that what is required to judge 
argument strength is an assessment of the 
degree to which the initial premisses render 
the final conclusion acceptable. Determin-
ing this numerically will involve prop-
agating acceptability through the support 
structure of the argument, beginning with 
the initial premisses, calculating their con-
tribution of acceptability to the subconclu-
sions, where these exist, and ultimately to 
the final conclusion. 

Thus I shall propose a method in which 
the assigned acceptability-values of the in-
itial premisses are used to calculate the 
values of what I shall call "received accep-
tability" for the remaining statements in the 
argument. The received acceptability of a 
conclusion mayor may not be equal to its 
real acceptability: received acceptability is 
an assessment of degree of support, not of 
real acceptability. It sets the lower limit of 
acceptability for a conclusion, but no more. 

I shall use the words "premiss" and 
"conclusion" to refer generally to the 
statements occurring in any inference-step 
in an argument, irrespective of whether they 
are initial premisses, final conclusions or 
subconclusions. For each inference-step, 
represented in the commonly-used tree 
diagram of the argument's structure by an 



22 John Black 

arrow, the support given by the premiss( es) 
to the conclusion is a function of the accep-
tability of the premiss(es) and the sufficien-
cy of the premiss(es) for the conclusion. In 
keeping with standard practice in informal 
logic, I shall use "acceptability" and "suf-
ficiency" in such a way that the sufficien-
cy of a premiss for a conclusion is indepen-
dent of the acceptability of the premiss, and 
that of the conclusion. 

C. Probability and Acceptability 

For the sake of this paper, I shall more 
or less identify these notions. 2 Both, of 
course, are open to a variety of interpreta-
tions and there are interesting parallels be-
tween disputes about each. Each notion can 
be treated as essentially objective, or as 
essentially subjective. These philosophical 
differences, however, do not impinge upon 
the applicability of the calculus in either 
case. I shall suppose that the calculus is 
equally applicable to acceptability and to 
probability, with one exception concerning 
conditional probability (see Section D). 

Acceptability can be either assigned or 
received. The former is an absolute value 
more or less equivalent to the probability 
of the relevant statement's being true. The 
latter is a value relative to a particular argu-
ment, and to a stage in such an argument. 
It is a measure of the accumulated support 
for a statement, the degree to which those 
statements which precede the statement in 
the support-structure of the argument render 
the statement acceptable. Both types of ac-
ceptability are equivalent in the way they 
can be used to calculate the received accep-
tability of statements which follow them in 
the argument-structure. 

The value of the acceptability of p is 
written A(P), in the case of assigned accep-
tability, and A(pl ... ) in the case of received 
acceptability, the lacuna being filled with 
a shorthand description of that from which 
p is inferred. This value ranges between 0 
and 1. Like probability, assigned accep-

tability can be determined for truth-
functions. Thus, 

(1) A(ip) == 1 - A(p) 

(2) A(p.q) == A(p) x A(q) 

(3) A(pvq) == A(p) + A(q) - A(p.q) 
= A(p) + A(q) 

(A(p) x A(q» by (2) 

(4) A(p-> q) = A (l(p. ~lq» 
== 1 A(p. iq) by (1) 
== 1 - (A(p) X 

A(iq» by (2) 
= 1 (A(p) x 

(1 - A(q))) by (1) 
== I - A(p) + (A(p) x A(q» 

Some qualifications need to be noted. 
First, (2) is true only when p and q are in-
dependent of each other, and when they are 
not mutually exclusive. (lfthey are mutually 
exclusive, A(p.q) == 0.) (2) is applicable, 
however, to all but the most contrived of 
examples, since the necessary conditions 
mentioned here are almost always fulfilled 
in any determination of the acceptability of 
a conjunction required by the present pro-
gramme. The latter determination is 
restricted to the cases of linked premisses 
and, in one way of understanding the 
transmission of acceptability, of the con-
junction of a premiss with the statement that 
the premiss entails a particular conclusion. 

The second qualification concerns (3). 
The subtraction of the product of the two 
acceptabilities from their sum is justified in 
the following way. In determining the prob-
ability to be assigned to a disjunction, sum-
ming the probabilities of the two disjuncts 
leads to counting twice those circumstances 
in which both disjuncts are true. They are 
counted once in the truth-conditions for each 
disjunct. The result is a value for the prob-
ability of the disjunction which is inflated 
by an amount equal to the probability of 
both disjuncts being true. This amount must 
then be subtracted from the sum of the 
probabilities of the disjuncts. Since I am 
here identifying assigned acceptability and 



probability, the same result follows for the 
former notion. 

Another way to look at this is to say that 
the assigned acceptability of the disjunction 
not being true is equal to that of both dis-
juncts being false. In other words, 

(5) A('(pvq» = A(lp. :q) 

(3) 

I - A(pvq) = A(:p) x AClq) 
by (1) and (2) 

A(pvq) = 1 - (A(lp) x A(iq» 
= 1 - «(1 A(p» x 

(l - A(q») by (I) 
1 (1 - A(p) -
A(q) + (A(p) x A(q») 

= A(p) + A(q) - (A(p) x 
A(q)) 

It will be seen later (in Section E3) that this 
feature solves a problem identified by Bat-
tersby in his original presentation in 
Amsterdam. 

As with (2), (3) applies only when p and q 
are independent of each other and not mutu-
ally exclusive; if they are mutually exclusive 
there is no need to perform the final subtrac~ 
~ion, ~ince then A(p.q) = O. Like (2), (3) 
IS applIcable to all serious real-life arguments. 

D. Acceptability and Sufficiency 

To calculate the acceptability transmit-
ted by a premiss to a conclusion (or, in other 
words, the received acceptability of the con-
clu~i?n, relative to that premiss), the accep-
tabllIt!' of the premiss is multiplied by its 
suffiCiency for the conclusion. The latter is 
a value, also ranging between 0 and I, writ-
ten as S(p for q), for premiss p and conclu-
sion q. The proposed way to handle the in-
itial assignment of sufficiencies is to iden-
tify them with acceptabilities. There are 
three candidates for a way in which this 
might be done, but only one is appropriate. 

The first is to use an analogue of con-
ditional probability: 

(6) S(p for q) :::: A(q,p) = A{qJi) ??? 
A(p) 

Quantifying Support 23 

This route is not plausible in the present 
case. It is not at all clear how one could 
calculate a value for A(q.p), since p and q 
are not likely to be independent. If one were 
allowed to employ (2), this would make 
A(q,p) equal to A(q), creating two major 
problems. First, A(q) is unknown-it is 
what is to be determined through the en-
~ire qu~n~ifi~ation process. Second, if per 
Impo~SlbIle It were known, the sufficiency 
of an mference would be identical to the ac-
ceptability of its conclusion, and this con-
tradicts the required independence of the 
sufficiency of an inference-step and the ac-
ceptability of its premiss and conclusion. 
In so far as it applies to the quantitative 
assessment of support, the calculus has no 
need of conditional probability, and I shall 
assume that there is no defined analogue in 
the present application. 3 

In~tead, I shall identify the sufficiency 
of an mference-step with the assigned ac-
ceptability of the corresponding conditional. 
!he second candidate method to be rejected 
Involves treating this conditional as a 
material conditional: 

(7) S(p for q) = A(p-> q) ??? 

This proposal also violates the required in-
de~e.ndence of s~fficiency from the accep-
tabJilty of premISS and conclusion. on ac-
count of the truth of (4). This leaves us with 
the third way, involving a stronger form of 
conditional, symbolized as: 

(8) S(p for q) = A(p--» q) 
When I talk of the conditional involved in 
the assessment of sufficiency, then, I mean 
it to be construed in terms of what I shall 
call "pragmatic implication. " I must leave 
this notion intuiti ve for the moment. I do 
not want a form of conditional that is 
restricted to deductively valid inferences; 
I do want an analogue of deductive entail-
ment which preserves one of its features 
but which can be applied to inferences of 
all kinds. 4 Just as whether or not p entails 
q, for any q, is independent of the truth of 
p. the sufficiency of p for any q is to be 



24 John Black 

independent of the acceptability of p. I shall 
refer to such conditionals in this context as 
"connecting conditionals." 

The substitution of the acceptability 
of a certain conditional for the sufficiency 
of an inference-step can be seen as 
equivalent to recasting the argument in 
deductively valid form. Corresponding to 
each inference-step is a conditional, which 
together with the premiss entails the con-
clusion. The acceptability transmitted to the 
conclusion is the acceptability of the con-
junction of the premiss and the conditional. 
Since premiss and conditional are mutual-
ly independent, this is equal to the product 
of their acceptabilities. 

E. How to Make the Numbers Work 

As I said above, the acceptability re-
ceived by q from the single premiss p is the 
product of the acceptability of p and the suf-
ficiency of p for q. Hence, 

(9) A(q/p) == A(p) x S(p for q) 
== A(p) x A(p--» q), by (8) 

or, more generally, 

(lO)A(q/p/ ... ) == A(p/ ... ) x S(p for q) 
== A(p/ ... ) x A(p--» q) 

What this means is that the acceptability 
received by q from p is equal to the 
acceptability of the conjunction of p and 
(p--» q). 

Apart from simple support, the calculus 
has to be able to handle serial, linked and 
convergent support. (In this context, 
divergent support is a trivial embellishment 
of simple support.) 

E 1. Serial Support 

For premiss p, subconclusion q and con-
clusion T, 

(11) A(r/q/p) == A(q/p) x 
Seq for r) by (9) 

== A(p) x S(p for q) 
x Seq for r) by (9) 

== A(p) x A(p--» q) 
x A(q--» r) by (8) 

or, generally, 

(12) ACr/q/p/ ... )== A(q/p/ ... ) x Seq for r) 
== A(p/ ... ) x S(p for q) 

x Seq for r) 
== A(p/ ... ) x A(p--» q) 

x A(q--» r) 

E2. Linked Support 

Linked support is handled by conjoin-
ing the premisses and calculating the pro-
duct of their acceptabilities and that of the 
connecting conditional whose antecedent is 
their conjunction. 

(13) S(p+q for r) == A«p.q)--» r) 

(14) A(r/p+q) = A(p) x A(q) x 
A«p.q)--» r) 

or, generally, 

(15) A(r/(p/ ... )+(q/ ... » == A(p/ ... ) x 
A(q/ ... ) x A«p.q)--» r) 

E3. Convergent Support 

The convergent case is slightly more 
complicated, and almost led Battersby to 
despair of his programme during the 
Amsterdam presentation. If a conclusion is 
supported by two independent premisses, 
we want to sum the contributions of each, 
and yet this can lead to acceptabilities 
greater than I, which contradicts our rules 
for assigning acceptabilities. The answer to 
the conundrum is provided, however, by the 
manner in which the probability calculus 
handles disjunction. In the simple case, the 
received acceptability of a conclusion r sup-
ported by two convergent premisses p and 
q is given by: 

(16) A(rlp;q) = A(r/p) + A(r/q) -
(A(r/p) x A(r/q» 

= (A(p) x A(p--» r» + 
(A(q) x A(q--» r» 
- «A(p) x A(p--» r» 
x (A(q) x A(q--» r») 

or, in the general case, 



(17) A(r/(p/ ... );(q/ ... » 
A(r/p/ ... ) + A(r/q/ ... ) (A(r/p/ ... ) 
x A(r/q/ ... » 
(A(p/ ... ) x A(p--» r» + 
(A(q/ ... ) x A(q--» r» 
- «A(p/ ... ) x A(p--» r» x 
(A(q/ ... ) x A(q--» r») 

This. equation indicates that the acceptability 
receIved by r from the two independent 
premisses p and q is equivalent to the 
acceptability of the disjunction of the con-
junctions p.(p--» r) and q.(q--» r). 
The subtraction of the product of the two 
contributions ensures that the acceptabili-
ty received by the conclusion never rises 
above 1. 

The same principle governs the formula 
for cases involving more than two con-
vergent premisses. In each case we calculate 
the acceptability of the disjunction of the 
conjunctions of each premiss with its con-
necting conditional. The precise 
mathematics are too detailed to be repeated 
here, but it can be seen that consideration 
ofthe process used to generate (3) from (5) 
yields: 

(18) A(r/(p/ ... ); ... ;(z/ ... » = 1 -
«1 A(r/p» x ...... x (1 - A(rlz») 

Before I move on, I wish to make a brief 
note about degrees of significance in 
relation to the calculation of received 
acceptability. The value calculated for the 
received acceptability of a conclusion 
should be stated to a number of significant 
figures no greater than the smallest number 
of significant figures occurring in the 
assignment of acceptabilities to premisses 
and connecting conditionals. This constraint 
is derived from some general considerations 
in error theory which it would be tedious 
to elaborate here. It is a moot point, for me, 
whether the rounding should take place only 
at the end of the propagation or at each 
stage in the argument-structure. I should 
recommend the first, but am unsure of the 
principles which might justify this 
recommendation. 

Quantifying Support 25 

F. Counter-considerations 

At Amsterdam, Battersby expressed the 
hope that any quantitative s)' .,tem developed 
for argument evaluation would be able to 
handle counter-considerations. The present 
one has that ability. To demonstrate this, 
I shall examine counter-considerations of 
two kinds: (1) those directed at a statement 
in the argument, and (2) those directed at 
an inference-step. 

Fl. Statement-counters 

Consider the simple case of a counter-
consideration q directed at a conclusion r. 
All that needs to be done in this case is to 
reduce the acceptability received by r from 
the rest of the argument up to that stage by 
an amount proportional to the acceptabili-
ty given by q to IT. If A(r/ ... ) is the in-
itial value calculated for the acceptability 
of r from the positive support for r, then 
the adjusted value is given by: 

(19) A(r/ ... \q) = A(r/ ... ) (A(r/ ... ) x 
A(lr/q» 

= A(r/ ... ) x (1 A(lr/q» 
A(lr/q) is of course given by: 

(20) A(lr/q) = A(q) x A(q--» r) 

Another way to understand (19) is to reflect 
that if the counter-consideration q were true, 
and if the conditional r were also 
true, then r could not be true. The support 
for r is thus contingent upon the falsity of 
the conjunction of the counter and its con-
necting conditional. To assess the value of 
that support, we need to consider the ac-
ceptability of the conjunction of the asser-
tion of the conjunction of the positive 
premiss and its connecting conditional 
t?gether with the negation of the conjunc-
tI?? of the counter and its connecting con-
dItIOnal. Take the simple case where the 
positi~e support for r comes from the single 
premISS p: 

(21) A(r/p\q) = A«p.(p--» r» 
I(q.(q--» r))) 



26 John Black 

= A(p.(p--» r» x 
A(i(q.(q--» r))) 

by (2) 
= A(p.(p--» r» x 

(1 A(q.(q--» r))) 
by (1) 

= A(r/p) x (1 A(ir/q)) 
by (9) 

The last line above can be generalized to 
yield (19). 

In more complicated cases, the value of 
AClq/p) may have to be calculated in com-
plicated ways, depending on whether there 
are many linked or convergent counter-
considerations, for example. The rule of 
thumb is to combine the counter-
considerations into one and then apply (19). 

F2. Inference-counters 

An argument may countenance a con-
sideration which seems to undermine not 
one of its statements, but instead one of its 
inferences. In this case, we need to adjust 
the value of the sufficiency accorded to the 
relevant inference-step. In the case of a 
counter q directed at the inference from p 
to T, 

(22) S(p for r\q) = S(p for r) (S(p for r) 
x AC1(p--» r)/q» 

or, by (8), 

(23) A(p--» r\q) = A(p--» r) 
(A(p--» r) x 
ACi(p--» r)/q» 

= A(p--» r) x (1 
A(i(p--» r)/q» 

where, 

(24) A(l(p--» r)/q) = A(q) x 
A(q--» -(p--» r» 

Once again, (23) can be read as saying that 
the acceptability of the connecting condi-
tional, in light of the counter-consideration, 
is equal to the acceptability of the conjunc-
tion of the connecting conditional with the 
negation of the conjunction of the counter 
and its connecting conditionaL 

Before we leave the topic of counter-
considerations, it is worthwhile to mention 
a general constraint on the assignment of 
values to the acceptability of counters. This 
is summarised, in the case of a counter p 
to a conclusion q, by: 

(25) A(q/ ... ) + A(lq/p) <:: 1 

This is so because the received accep-
tabilities of each of q and -;q are the lower 
bounds of their real acceptabilities, and the 
latter sum to 1. 

G. Thomas's Adjectives 

As I mentioned earlier, Stephen Thomas 
also made a presentation in Amsterdam. He 
argued that his system of assessing argu-
ment strength had proved to be successful 
in classroom trials, and had enabled his 
students to evaluate many of Hume's 
arguments, for example, without prior ex-
posure. Thomas's method applies, so far, 
only to single inferences, whether simple 
or linked. Like the method proposed here, 
it also involves combining an assessment 
of acceptability with one of sufficiency to 
reach one of argument "soundness and 
reliability." Unlike those employed in the 
present quantitative method, these assess-
ments are not expressed in numerical terms. 

According to material distributed by 
Thomas at Amsterdam, what I call accep-
tability is evaluated using the terms 
"definitely true," "probably true," 
"uncertain/don't know," "probably false" 
and "definitely false." What I call suffi-
ciency (Thomas prefers "validity") is ex-
pressed using the terms "deductively 
valid," "strong," "moderate," "can't 
tell," "weak" and "nil." The strength of 
the argument, or reasoning, is expressed us-
ing the terms "S: sound and reliable," "PS: 
probably sound and reliable," "M: 
marginal" ( = close to sound and reliable, 
but not quite certain or totally reliable 
beyond a reasonable doubt), "CS: cannot 
say," "PU: probably unsound and 



unreliable" and "U: unsound and 
unreliable." For brevity, I shall use 
"sound" instead of "sound and reliable, " 
and so forth. 

I propose to test Thomas's intuitions 
about degrees of argument strength by us-
ing part of my method, the part which in-
volves multiplying acceptability-values by 
sufficiency-values to reach a numerical 
assessment of argument strength. (These in-
tuitions, of course, mayor may not accord 
with my own or with the reader's.) To do 
this I'll use a "spectrum of validity" 
diagram provided by Thomas at Amsterdam 
(a similar diagram appears on p. 135 of his 
textbook mentioned above) to assign values 
of sufficiency (or validity, in his ter-
minology), as follows: 

Deductively valid = 1 
Strong = 0.95 - 0.99 
Can't tell (rare) = 0.5 
Moderate = 0.53 - 0.94 
Weak = 0.01 - 0.52 
Nil = 0 

These values, aside from the first, third and 
last, are based on a linear measure of 
Thomas's diagram. The diagram is rough: 
it is possible that Thomas intended the 
border between "moderate" and "weak" 
to fall at 0.5, for example, but otherwise 
the level of accuracy is sufficient for my 
purposes. The first and last values are ob-
vious; the third just gives the best estimate 
in a case where nothing inclines one either 
way on the judgment of sufficiency. 

Assigning acceptability values is a bit 
more complex, but here goes: 

Definitely true = 1 
Probably true = 0.75 - 0.99 
Don't know = 0.26 - 0.74 
Probably false = 0.01 0.25 
Definitely false = 0 

The problems here are the "probably" 
statements. Thomas defines "probably 
true/false" as "it is highly likely that this 
statement is true/false, but I do not know 
this with complete certainty." I was 

Quantifying Support 27 

tempted to assign ranges identical to that 
for "strong," by analogy with his spectrum 
for validity, where, for example, he defines 
"strong" as: 

If the reasons were true, they would make 
the truth of the conclusion extremely like-
ly, certain beyond any reasonable doubt. 
"virtually a sure thing," but not totally 
guaranteed. Likely enough to make it 
reasonable to stake something of great value 
on the truth of the conclusion if the reasons 
are true, and likely enough to serve as a 
definitely reliable basis for actions. 5 

It's arguable, though, that this second 
definition is more restrictive than the first-
though not entirely so. I have thus extend-
ed the ranges. I also wished to ensure that 
there are no gaps in the scheme for accep-
tability, Hence the broad range for' 'don't 
know, " 

For the sake of argument, I shall take 
values in the middle of each range as 
representative of the range, Thomas pro-
vides the following matrix for judgments 
of argument strength, I have amended it by 
writing in the values for premiss-
acceptability, inference-sufficiency and 
argument strength, as determined by their 
product, the received acceptability of the 
conclusion: 

TABLE OF SOUNDNESS AND 
RELIABILITY VERDICTS5 

DEGREE OF VALIDITY 
(DEGREE OF CONFIRMATION 

OF THE INFERENCE) 

STATUS Oed Strong Can't Mod- Weak 
OF THE Valid Tell crate 

REASONS I 0.97 0,5 0.74 0.26 

Del' S S CS U U 
true I I 0.97 0.5 0.74 0.26 

Prob PS M CS U U 
true 0,87 0.87 0.84 0.44 0.64 0.23 

Don't CS CS CS U U 
know 0.5 0.5 0.49 0.25 0,37 0.13 

Prob PU PU 
t~7 

U U 
false 0.13 0.13 0.13 0,1 0.03 

Del' U U U U U 
false 0 0 0 0 0 0 

Nil 

0 

U 
0 

U 
0 

U 
0 

U 
0 

U 
0 



28 John Black 

Examination of the numerical matrix 
reveals that in several areas there is a clash 
of intuitions. Of course this may be due to 
the artificiality of my assignments of values 
to Thomas's adjectives, especially in the 
area of acceptability, but it may also be a 
result of genuine differences in subjective 
estimates of argument strength. 

The first area of concern is the estimate 
of argument strength when the premisses 
are either definitely true or probably true 
and the reasoning is moderately valid. My 
method (in company with my intuitions) in-
dicates that Thomas's assessment of "un-
sound" is rather harsh, since the correspond-
ing values of received acceptability for the 
conclusion are 0.74 and 0.64. Whatever 
value we take within the range assigned to 
a moderate degree of validity, the argument 
strength would be greater than 0.5 in all 
cases when the premisses are definitely true, 
and in many when they are merely probably 
true. In many cases the estimate of argu-
ment strength in these categories will be 
greater than 0.9, given that moderate validi-
ty extends up to sufficiency-values of 0.94. 
I think Thomas is a more conservative 
gambler than I! 

I should, however, stand behind my in-
tuitions in these cases. Thomas's verbal 
description of moderate validity is: 

Less than "strong" but more than "weak." 
If the reasons were true, they would not 
establish the truth of the conclusion as "a 
sure thing," but they would at least make 
it a "good bet." However, more would be 
required to establish the conclusion beyond 
any reasonable doubt. 5 

Thomas's table of soundness verdicts seems 
a little at odds with this description of the 
acceptability of the conclusion in the case 
when the premisses are true, and in fact this 
description seems more in accord with the 
values assigned by my method. ! 

The second area of concern is the one 
in which the premisses are probably false, 
though the reasoning is deductively valid, 
strong or unknown. In these cases, Thomas's 
judgments of "probably unsound" seem 

rather lenient when compared with the 
numerical values generated by my method. 
This impression is magnified when a com-
parison is made with the first area of con-
cern. I suggest that Thomas's intuitions are 
influenced to an inappropriate degree by the 
validity-factor and insufficiently by the 
premiss-factor. Could they be the intuitions 
of one who, like me, has devoted many 
years to trying to get students to 
acknowledge the importance of the concepts 
of inference-strength (validity, sufficiency) 
to the evaluation of arguments? My method, 
based heavily on acceptability, kowtows to 
the undergraduate fixation with issues of 
truth and falsity, but as a result produces 
what I think is a more balanced table of 
soundness verdicts. 

The final area of concern is the verdict 
when the premisses are probably true 
("highly likely") and the reasoning is 
strong. Thomas's soundness verdict of 
"marginal" seems a little harsh when one 
considers that the only form of validity 
which is stronger than "strong" is deduc-
tive validity, and "probably true" includes 
all strong likelihoods except absolute cer-
tainty. The value assigned by my method 
reflects my intuition that the reasoning is 
at least "probably sound." 

Thus I should be prepared to defend my 
method against the apparent deviations from 
Thomas's intuitions which it necessitates. 
I should remind the reader once again, 
however, that the initial assignment of 
values upon which the consideration of 
Thomas's table is based is provisional to a 
great extent. I am sure that refinements 
could be suggested. Nevertheless, I think 
that the approach I have taken is vindicated 
by a comparison with Thomas's parallel but 
non-quantitative approach. The comparison 
suggests to me that the precision of the 
numerical assessment of argument strength 
provides a certain level of objectivity. In-
herent in the relative imprecision of the ad-
jectival approach is a temptation to subjec-
tivity, for which my method can be seen as 
an antidote. 



H. Reservations 

Having spent some time developing and 
motivating the above method for the quan-
titative assessment of argument strength, I 
now wish to record my strong reservations 
about its general applicability. The problem 
lies, I think, not in the calculation of the 
way in which acceptability can be pro-
pagated through an argument's structure, 
but in the initial assignment of acceptability-
values both to premisses and to connecting 
conditionals. 

Certain assignments are non-
controversial and these, I suppose, 
guarantee the method some application, 
even if a limited one. Included in this set 
of assignments will be cases of known truth 
and falsity (values of lor 0), though in these 
cases the use of the calculus method will 
likely be both trivial and burdensome. Also 
included will be the application to 
probability-statements, where fractional 
values gain a foothold. Arguments all of 
whose premisses and connecting condi-
tionals fall into these two classes of state-
ment will then be non-controversially 
evaluable using the quantitative method. 

I am not confident, however, that it will 
be applicable in any other cases. The dif-
ficulty of assigning "accurate" values to 
the acceptability of statements in general is 
not an epistemic difficulty, but a concep-
tual one. What does it mean to say that the 
acceptability-value of "Brian Mulroney's 
days in office are numbered" is 0.93? Not 
a lot, it seems to me. At most, the numerical 
value could be treated as an estimate of the 
SUbjective degree of confidence which the 
argument-evaluator holds in the truth of the 
statement. Here it is hard to see that it makes 
sense to specify the value beyond the first 
significant figure-our intuitions are just not 

Quantifying Support 29 

that fine-grained, which is why Thomas's 
method is far more intuitively appealing. 

Still, it might be thought, even this 
limited approach might be useful in testing 
and correcting our unquantified intuitions 
about argument strength. I have used the 
method to this end in examining Thomas's 
table of soundness verdicts. Employed in 
this way, the quantitative method seems to 
be a useful tool in the workshop of argu-
ment evaluation. Its use, however, is largely 
negative: I think it would seldom, if ever, 
be plausible to prefer the quantitative result 
to one's own thought-out intuitions. 

In any real case of conflict between in-
tuitive and quantitative assessments of argu-
ment strength, I would be inclined to let my 
intuitions about argument strength overrule 
the result dictated by the quantitative 
method. Rather than admit that my intui-
tions were wrong, I would go back and alter 
the initial assignments of acceptability to the 
premisses or connecting conditionals so that 
the result comes out more in line with my 
intuitions. This is not stubbornness-the 
point is that I am likely to be more certain 
in my estimate of argument strength than 
I am in my assignment of initial accep-
tabilities. (My attitude here is parallel to the 
one I should evince if a moral theory which 
I thought I held dictated an intuitively un-
palatable verdict on a specific case of moral 
action.) 

My conclusion, then, is a modest one: 
to the limited extent that a quantitative 
method for assessing argument strength is 
viable, it should take the form described 
herein. Having spent some effort in 
developing the method, I should be 
delighted if someone could suggest a class 
of applications for which it is really useful, 
but that task I must leave to minds of a bent 
more formal than my own. 

Notes 

1 Stephen Thomas. Practical Reasoning in 
Natural Language, Englewood Cliffs. 
Prentice-Hall, 1986. 

2 In his book Plausible Reasoning, Amsterdam. 
Van Gorcum, 1976, Nicholas Rescher in-
vestigates in a quasi-quantitative way the 



30 John Black 

notion of plausibility. Some passages appear to 
suggest, though this is not entirely clear, that 
he thinks the latter notion is to be preferred to 
that of probability in assessing the acceptabili-
ty of a conclusion drawn from its premisses. 
Space does not allow for a full treatment of this 
interesting claim, but I shall offer one comment 
here. It is a feature of Rescher's account of 
plausibility that a conclusion (of a deductive 
argument) is no more and no less plausible than 
its least plausible premiss; this distinguishes 
plausibility from probability. It has the conse-
quence, however, that a conclusion gains no in-
crease in plausibility when a premiss other than 
the least plausible of the premisses is discovered 
to be more plausible than originally thought. 
In my view, this feature prevents conclusion-
plausibility from being considered an adequate 
assessment of argument strength, as premiss-
plausibility from being an adequate assessment 
of premiss-acceptability. For if Rescher's ac-
count were applicable to the case imagined here, 
an increase in premiss-acceptability would pro-
duce no increase in argument strength. 

3 This may be unduly pessimistic. At the very 
least it might be possible to employ the notion 
of conditional acceptability, defined in terms 
of real acceptabilities, for purposes other than 
the definition of sufficiency. For an investiga-
tion of similar possibilities in relation to the 
special case of plausible reasoning in a deduc-
tive context, see George Polya, Mathematics 
and Plausible Reasoning, Princeton, Princeton 
University Press, 1954. Polya's notion of 
plausible reasoning differs from Rescher's (see 
previous note). Polya is concerned with the in-
crease in the plausibility of a conjecture when 
one of its deductive consequences is found to 
be true. 

4 For similar declarations about this important no-
tion, see Jonathan Berg, "Interpreting 
Arguments," Informal Logic IX.I, 1987, and 
Trudy Govier, A Practical Study of Argument, 
Belmont, Wadsworth, 1985, p. 60 ff. 

5 These passages are all taken from material 
handed out by Thomas at his presentation in 
Amsterdam. Similar fommlations occur in his 
textbook. 

Appendix: Explanation of Symbols 

-'p = the negation of p = not-p A(plq;r) = the received acceptability of p, 
p.q the conjunction of p and q relative to convergent premisses 
pvq the disjunction of p and q = p or q q and r 
p-0 q = if P then q (material conditional) A(p\q) the assigned acceptability of p, 
p--»q if p then q (pragmatic implication) despite counter-consideration q 

A(plq\r) = the received acceptability of p, 
A(p) the assigned acceptability of P relative to q, despite counter-
A(p/q) the received acceptability of p, consideration r 

relati ve to q 
S(p for q) = the sufficiency of p for q A(p/ ... ) the received acceptability of p, S(p for q\ r) = the sufficiency of p for q, despite 

relative to whatever precedes p in counter-consideration r 
the argument-structure 

A(p/q/r) the received acceptability of p, JOHN A. BLACK 
relative to q, which is preceded in PHILOSOPHY DEPARTMENT 
the argument-structure by r MALASPINA COLLEGE 

A(p/q+r) = the received acceptability of p, 900 FIFTH STREET 
relative to linked premisses q and r NANAlMO, B.C. V9R 5S5 0